Problem: Let $f(x)=-8\cdot7^x$. Find $f'(x)$. Choose 1 answer: Choose 1 answer: (Choice A) A $-8\cdot 7^x\ln(7)$ (Choice B) B $-8\cdot 7^x\log_7(x)$ (Choice C) C $-8\cdot 7^{x-1}$ (Choice D) D $-56^{x-1}$
Solution: The expression for $f(x)$ includes an exponential term. Remember that the derivative of the general exponential term $a^x$ (where $a$ is any positive constant) is $\ln(a)\cdot a^x$. Put another way, $\dfrac{d}{dx}(a^x)=\ln(a)\cdot a^x$. $\begin{aligned} f'(x)&=\dfrac{d}{dx}(-8\cdot7^x) \\\\ &=-8\dfrac{d}{dx}(7^x) \\\\ &=-8\cdot\ln(7)\cdot7^x \\\\ &=-8\cdot7^x\ln(7) \end{aligned}$ In conclusion, $f'(x)=-8\cdot7^x\ln(7)$.